Thesis Crack Modelling With the EXtended Finite Element Method

Crack Modelling with the eXtended Finite Element Method

Francisco Xavier Girão Zenóglio de Oliveira

Thesis to obtain the Master of Science Degree in Aerospace Engineering

Examination Committee Chairperson: Prof. Fernando José Parracho Lau Supervisor: Prof. Virgínia Isabel Monteiro Nabais Infante Co-supervisor: Prof. Ricardo Miguel Gomes Simões Baptista Members of the Committee: Prof. Ricardo António Lamberto Duarte Cláudio Prof. José Miguel Almeida da Silva

July 2013

II

Dedicated to my family and friends

III

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Agradecimentos Deixo aqui o meu especial agradecimento a` senhora professora Virg´ınia Isabel Monteiro Nabais ˜ Infante, e ao senhor professor Ricardo Miguel Gomes Simoes Baptista, que sempre se revelaram ˜ desta tese. dispon´ıveis na realizac¸ao ´ a` minha fam´ılia, que me tem sempre apoiado ao longo da minha vida academica. ´ Agradec¸o tambem ´ Por ultimo, a todos os meus amigos, de secundario, bem como aqueles que conheci nesta casa, ´ Instituto Superior Tecnico, que sempre me mostraram que a vida e´ mais do que somente o nosso trabalho.

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Acknowledgements I leave here my special thanks to professor Virg´ınia Isabel Monteiro Nabais Infante, and professor ˜ Ricardo Miguel Gomes Simoes Baptista, who always proved to be available in the realization of this thesis. I also thank to my family, who has always supported me through my academic life. Finally, to all my friends from high school, as well as those I met in this house, Instituto Superior ´ Tecnico, who always showed me that life is more than just work.

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Resumo O mais importante para a industria aeroespacial e´ a seguranc¸a do equipamento. Os engenheiros ´ ˜ de qualidade. O estudo de fendas e´ extremafazem um grande esforc¸o para garantir elevados padroes ´ mente importante para este proposito. ˜ de fendas, tem sido desde sempre um topico ´ A modelac¸ao muito importante. As abordagens tradi´ ˜ precisas, no entanto, a construc¸ao ˜ cionais com o metodo dos elementos finitos podem fornecer soluc¸oes ˜ e´ obvia. ´ das malhas e´ demorada e nao ´ Um novo conceito emerge, conhecido como o Metodo de Elementos Finitos Extendidos, XFEM, ´ ˜ introduzidas numericamente, com a em que as descontinuidades geometricas e as sigularidades, sao ˜ de novos termos as ` func¸oes ˜ de forma. Assim, a formulac¸ao ˜ em elementos finitos permanece a adic¸ao ˜ da fenda e´ mais facil, ´ ˜ mais precisa. mesma, a representac¸ao com uma soluc¸ao ´ Esta tese verifica a validade deste novo conceito para fendas estacionarias com ajuda do XFEM, R ´ ˜ e´ o factor de intensidade de tensoes ˜ implementado no Abaqus . O criterio de comparac¸ao para ge-

˜ proximos ´ ˜ ometrias simples. Os resultados computacionais sao dos valores obtidos com as soluc¸oes ´ dispon´ıveis na literatura e qualitativamente a simplicidade do metodo e´ verificada.

Palavras-chave: Fenda, XFEM, Abaqus R

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X

Abstract The most important thing for the aerospace industry is the equipment’s safety. Engineers, make a great effort to guarantee high standards of quality. The study of crack phenomena is major for this purpose. Crack modelling, has ever been an important topic. Traditional approaches of the finite element method can provide accurate solutions, nevertheless the meshing techniques are time consuming and not obvious. A new concept emerges, known as the eXtended Finite Element Method, XFEM, where the geometric discontinuities and singularities, are introduced numerically with the addition of new terms to the classical shape functions. So, the finite element formulation remains the same, the crack representation is easier, with an approximate solution more precise. R This thesis, verifies the validity of this new concept for stationary cracks with Abaqus ’s XFEM aid.

The comparison criterion is the stress intensity factor for simple geometries. The computational results are near to the values obtained from the closed-forms available on the literature and qualitatively the simplicity of this method is checked.

Keywords: Crack, XFEM, Abaqus R

XI

XII

Table of Contents Agradecimentos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements

V

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XI

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII 1 Introduction

1

1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.4 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 Bibliographic Research

5

2.1 Theory of Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.1 Stress Distribution Around a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1.2 Loading Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.3 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.4 The Griffith Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.5 The Energy Release Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.1.6 The J-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2.1 System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2.2 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.2.3 Element Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.3 The Classical Approach to the Stress Intensity Factor Calculation . . . . . . . . . . . . . .

19

2.4 The eXtended Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4.1 XFEM Enrichment: Jump Function . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.4.2 XFEM Enrichment: Asymptotic Near-Tip Singularity Functions . . . . . . . . . . .

23

XIII

2.4.3 XFEM Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Case Studies

23 25

3.1 The SENT Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.2 The CCT Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.3 The SENB Specimen

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.4 Cylindrical Pressure Vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

4 Numerical Study

33

4.1 The SENT Specimen Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.1.1 Mesh Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.1.2 Requested Contours and GRef Influence . . . . . . . . . . . . . . . . . . . . . . .

36

4.1.3 Influence of the Ratio a/W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.1.4 DRef Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.1.5 Interpolation and Integration

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.1.6 Standard Element Size Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.1.7 SENT Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.1.8 SENT Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.2 The CCT Specimen Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.2.1 Mesh Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.2.2 GRef Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.2.3 Standard Element Size Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.2.4 CCT Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.3 The SENB Specimen Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.3.1 Mesh Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.3.2 GRef Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.3.3 Standard Element Size Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.3.4 SENB Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.4 Vessel Under Pressure, Closed-Form Deduction . . . . . . . . . . . . . . . . . . . . . . .

59

4.4.1 The Geometry, Mesh Construction and Boundaries Conditions . . . . . . . . . . .

59

4.4.2 Results And Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5 Conclusions

65

5.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Bibliography

68

XIV

List of Tables 4.1 Impact in the average error of the requested number of contours . . . . . . . . . . . . . .

37

4.2 SENT analyses characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.3 Numerical study, files appearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.4 SENT GRef influence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.5 Ratio a/W influence for GRef =80 divisions . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.6 SENT DRef influence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.7 C3D8R versus C3D4R analyses characteristics

. . . . . . . . . . . . . . . . . . . . . . .

43

4.8 Integration effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.9 Standard element size attribution SENT results . . . . . . . . . . . . . . . . . . . . . . . .

46

4.10 SENT classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.11 CCT analyses characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.12 CCT GRef influence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.13 Standard element size attribution CTT results . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.14 CCT classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.15 SENB analyses characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.16 SENB GRef influence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.17 Standard element size attribution SENB results . . . . . . . . . . . . . . . . . . . . . . . .

57

4.18 SENB classical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.19 Vessel convergence evaluation, analyses characteristics . . . . . . . . . . . . . . . . . . .

60

4.20 Vessel convergence evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.21 Vessel results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

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XVI

List of Figures 2.1 Real and ideal crack tension behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2 Stresses near the crack tip and polar coordinates . . . . . . . . . . . . . . . . . . . . . . .

6

2.3 Load modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.4 A through-thickness crack in an infinitely wide plate subjected to a remote tensile stress .

9

2.5 J-Integral, 2D contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.6 3D J-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.7 Finite element domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

R elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Abaqus

17

2.9 Degeneration of a quadrilateral element into a triangle at the crack tip . . . . . . . . . . .

19

2.10 Degeneration of a brick element into a wedge . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.11 Crack-tip elements for elastic and elastic-plastic analyses . . . . . . . . . . . . . . . . . .

20

2.12 Spider-web mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.13 XFEM deduction, first mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.14 XFEM deduction, final mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

R 2.15 Abaqus enrichment scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.1 Edge crack in a semi-infinite plate subject to a remote tensile stress . . . . . . . . . . . .

26

3.2 The SENT specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.3 The CCT specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.4 The SENB specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.5 Broken pressure vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.6 Vessel with semicircular crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.7 Vessel geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

4.1 Perfectly structured mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.2 Unstructured mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.3 The SENT specimen final mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.4 Mesh refinement control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4.5 Error evolution for different number of requested contours . . . . . . . . . . . . . . . . . .

37

4.6 Time evolution with the GRef parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4.7 SENT contour error evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

XVII

4.8 Error behaviour versus a/W ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.9 DRef influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.10 Stresses comparison between the two kinds of integration . . . . . . . . . . . . . . . . . .

45

4.11 Standard element size attribution, SENT absolute average error evolution . . . . . . . . .

46

4.12 Classical partition scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.13 SENT, Classical mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.14 CTT mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.15 CCT contour error evolution without GRef = 10 . . . . . . . . . . . . . . . . . . . . . . . .

52

4.16 Standard element size attribution, CTT absolute average error evolution . . . . . . . . . .

53

4.17 CCT classical partition scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.18 CCT Classical mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.19 SENB partition scheme and mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.20 SENB mesh detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.21 SENB contour error evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.22 Standard element size attribution, SENB absolute average error evolution . . . . . . . . .

57

4.23 SENB classical partition scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.24 SENB Classical mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.25 Vessel geometry with boundary conditions and internal pressure . . . . . . . . . . . . . .

59

4.26 Vessel lateral view with semicircular crack . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.27 Vessel radial stress distributions for different ESize values . . . . . . . . . . . . . . . . . .

60

4.28 Vessel geometric factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

XVIII

Abbreviations R C3D4 Abaqus Linear Tetrahedral Element with 4 nodes

R C3D8 Abaqus Linear Hexahedral Element with 8 nodes

R C3D10 Abaqus Quadratic Tetrahedral Element with 10 nodes

R C3D20 Abaqus Quadratic Hexahedral Element with 20 nodes

CAE Computed Assisted Engineering CCT Centre Crack Tension DOF Degrees of Freedom DRef Depth Refinement ESize Element Size for the Standard Element Size Attribution FEM Finite Element Method GRef Global Refinement LEFM Linear Elastic Fracture Mechanics SENB Single Edge Notch Bending SENT Single Edge Notched Tension XFEM eXtended Finite Element Method

XIX

XX

Nomenclature 

Strain

Γ

Contour containing the crack tip

γs

Surface energy

µ

Shear modulus

ν

Poisson coefficient

Ω

Finite domain

Φ

Total energy

Π

Potential energy

Ψi

Crack tip asymptotic solutions

σf

Fracture stress

σh

Hoop stress

σij

Stress tensor

σrr

Radial stress

θ

Polar coordinate, angle

A

Crack area

a

Crack length

ai

Vector of enriched nodes with the Jump function

Am

Amplitude, dimensionless function of θ

B

Thickness

bi

Vector of enriched nodes with crack tip asymptotic solutions XXI

C

Closed contour

E

Young’s modulus

f

Volume forces

fij

Dimensionless function of θ in the leading term

G

Energy release rate

H

Jump function

I

Identity matrix

i

Number of the Depth Refinement

J

J-Integral

j

Number of requested contours

K

Stress Intensity Factor

k

Constant

Kc

Fracture toughness

Ki

Stress intensity factor, i mode

m

Interior normal

n

Exterior normal

Ni

Shape functions

P

Pre-logarithmic, energy factor tensor

pint

Vessel internal pressure

q

Vector within the virtual displacement

r

Polar coordinate, radius

ro

Outer radius

rp

Plastic radius

ri

Inner radius

S

Contour area XXII

t

Tension forces

ui

Displacement on the i direction

W

Elastic strain energy

Ws

Work necessary to create new crack faces

Y (a)

Geometric factor, function of the crack length

XXIII

XXIV

Chapter 1

Introduction 1.1

Context

Nowadays, the study and evaluation of the integrity of the various mechanical components is extremely important for the industry. The two major goals are: increase the components fatigue life expectancy, and their safety. Previously, the study and analyses were constantly made from laboratory experiments, which were lengthy, expensive [1] and sometimes difficult to implement, due to several rules [2]. Today, there are new materials and mechanical components appearing to a daily rhythm, which must be tested and certified to be available for the consumers. Inevitably, the engineers presented with systematic solutions, with higher precision, such as the Finite Element Method, making the whole process effective and efficient. However, some mechanical phenomena have always been difficult to model with the Finite Element Method (FEM), amongst which the study of cracks, stationary and especially its propagation. This particular study is necessary in order to predict the mechanical behaviour of the equipment but also in order to increase its life expectancy. Thus, there has always been a great need of representing correctly cracks, with accurate results and an expeditious manner. There were several attempts, which proved to be difficult to enforce for several reasons, the main one being the mesh generation around the crack tip [1] to obtain good results. It is in these terms that a new concept emerges recently; XFEM, eXtended Finite Element Method, creating a new paradigm in the study of cracks [3, 4]. The method of extended elements permits a representation of cracks by finite elements, which does not require to change the mesh to monitor crack propagation [5], causing a revolution when compared with the classical methods. The discontinuity in the elements is described from enrichments functions overlapping the elements. 1

1.2

Objectives

The XFEM, although a relatively recent concept, 1999, is now available in commercial versions of finite elements. In fact, the XFEM has been already implemented in the Finite Element Method software R Abaqus / Standard, owned by Simulia [6].

R A need therefore arises of realizing the potential and the Abaqus ’s XFEM validity in order to un-

derstand the possibilities for subsequent analyses. The aim is thus to evaluate the XFEM for crack R modelling in Abaqus .

The basic idea behind the theme is to understand how accurate is the increase brought by the XFEM in the fracture mechanics domain, to predict the integrity of mechanical systems, according to standard methods. The main objective is to conduct a convergence analysis of the FEM discretizations, identifying aspects and important parameters in the cases studied. The understanding of which are the best modelling techniques with the aid of XFEM, to achieve the best precision and efficiency within the mechanical behaviour of materials, will be the main objectives of this thesis. So, some simulations will be performed, with solutions known by the scientific community, R using Abaqus ’s XFEM.

1.3

Method

As first instance, it is made a bibliographic research to understand the principles of the linear elastic fracture mechanics theory, in order to explain the calculation of the stress intensity factor; the main parameter under analysis in this thesis. Then, the fundamentals behind the XFEM are also briefly presented. R Next, it is done an evaluation of the XFEM in Abaqus , where for a given geometry, with a solution

available in the literature for its stress intensity factor, it is studied the advantages and disadvantages of using the XFEM for different meshes, elements, rules of integration and interpolation. Furthermore, a better validation of the XFEM is achieved with two others geometries, to verify and confirm the conclusions from the first geometry analysis, to once again investigate the quality and applicability of XFEM. Finally, a more complex geometry is used in order to achieve a better understating of the XFEM limitations, allowing this thesis conclusion. 2

1.4

Structure

Chapter 2 discusses initially the basic concepts of the Linear Elastic Fracture Mechanics, where it can be understood how to calculate the stress intensity factor. In the second phase, the traditional finite elements and their constitutive laws are presented, allowing after the introduction of the XFEM, where its formulation is also presented. Chapter 3 makes a brief presentation of the geometries under analysis. Their geometric characteristics are briefly commented, and the closed-forms for theirs stress intensity factors are introduced. In chapter 4, all the numerical analyses are carried. First, several analyses are preformed. In the end the more important are identified in order to be better investigated with the other geometries. Finally, a more complex structure, consisting of a vessel under internal pressure, is analysed in order to conclude about the XFEM use for structures analysis. Chapter 5, presents the mains conclusion of this thesis, and suggest some further works.

3

4

Chapter 2

Bibliographic Research 2.1 2.1.1

Theory of Linear Elastic Fracture Mechanics Stress Distribution Around a Crack

The cracks in mechanical components subject to applied loads behave very close to what is observed when there are notches, which are responsible for stress concentration due to reduction of area against the nominal area. The geometry of the crack creates high stress concentrations in its tip. This behaviour is illustrated in figure 2.1. Due to the high tension observable on the edge of the crack, a plastic zone appears. However, following the LEFM theory, the plastic behaviour is not taken into account, and tension is given by an ideal crack following the linear elastic model. Consequently, the LEFM reveals a large gap, by not taking into account areas that could be in the plastic domain [7].

Figure 2.1: Real and ideal crack tension behaviour [7] Consulting Anderson [7], for cracked geometries subjected to external forces, it is possible to derive closed-forms or analytical expressions for the stresses in the body, assuming the LEFM. Irwin [8], 5

Sneddon [9], Westergaard [10], and Williams [11] were among the first to publish such solutions.

Figure 2.2: Stresses near the crack tip and polar coordinates [7]

Considering a polar coordinate system, (r, θ), with the origin at the crack tip represented on figure 2.2, it can be shown that the stress field in any linear elastic cracked body is given by,

∞ X k (m) σij = ( √ )fij (θ) + Am rm/2 gij (θ) r m=0

(2.1)

Where σij is the stress tensor, k a constant and fij a dimensionless function of θ in the leading term. For the higher-order terms, Am is the amplitude and a dimensionless function of θ for the mth term. It should be noticed, that the solution for any given configuration contains a leading term that is √ proportional to 1/ r. As r → 0, the leading term goes to infinity, but the other terms remain finite or √ approach zero. Thus, stress near the crack tip varies with 1/ r, independently of the geometries. It can √ also be shown that displacement near the crack tip varies with r. Equation 2.1 describes also a stress singularity, since stress is asymptotic to r = 0.

2.1.2

Loading Modes

In fracture mechanics, there are three types of loading that a crack can experience, presented on figure 2.3. Mode I loading, where the principal load is applied in the normal direction to the crack plane, opening the crack (a traction mode). Mode II corresponds to an in-plane shear loading and tends to slide one crack face over the other (a shear mode). Mode III refers to the out-of-plane shear (a torsion mode). A cracked body can be loaded in any one of these modes, or a combination of two or three modes. For each of these modes, it can be deduced a stress intensity factor, which are presented next. 6

Figure 2.3: Loading modes I, II and III [7]

2.1.3

Stress Intensity Factors

The stress intensity factors are used as a measure that quantifies the severity of a crack relatively to others cracks [7]. They are so, of extreme importance for the cracks study. They are also related to the mechanisms of crack initialization but also their propagation, and in some cases, the stress intensity factor may reach an extreme value: the fracture toughness KC , leading to the fracture of the components. Each mode of loading produces the

√1 r

singularity at the crack’s tip, but the proportionality constants

k and fij depend on the mode. For further considerations it is important to substitute k of equation 2.1 √ by the stress intensity factor K, where K = k 2π . The stress intensity factor is usually given with a subscript to denote the mode of loading, i.e., KI , KII , or KIII . Considering the LEFM, the stress fields ahead of a crack tip in an isotropic linear elastic material can be written as,

KI (I) (I) limr→0 σij = √ fij (θ) 2πr

(II)

limr→0 σij

(III)

limr→0 σij

(2.2)

KII (II) =√ fij (θ) 2πr

(2.3)

KIII (III) fij (θ) =√ 2πr

(2.4)

For modes I, II, and III, respectively. In a mixed-mode problem (i.e., when more than one loading mode is present), the individual contributions to a given stress can be summed:

(T otal)

σij

(I)

(II)

= σij + σij

(III)

+ σij

Equation 2.5 results from the principle of linear superposition. 7

(2.5)

Considering this thesis will focus on loading Mode I, it is shown below both the stress and displacement field ahead a crack tip,

  θ KI 3θ cos(θ) 1 − sin( )sin( ) =√ 2 2 2πr

(2.6)

  θ θ 3θ KI cos( ) 1 + sin( )sin( ) σyy = √ 2 2 2 2πr

(2.7)

KI θ θ 3θ τxy = √ cos( )sin( )cos( ) 2 2 2 2πr

(2.8)

σzz = 0

(2.9)

σzz = ν(σxx + σyy )

(2.10)

σxx

For plane stress,

For plane strain,

Where (r, θ) are the polar coordinates, ν the Poisson ratio. The remaining components of the stress tensor are zero. For the displacement field,

KI ux = 2µ

r

KI uy = 2µ

r

  θ r 2 θ cos( ) κ − 1 + 2sin ( ) 2π 2 2

(2.11)

  r θ 2 θ sin( ) κ + 1 − 2cos ( ) 2π 2 2

(2.12)

Where u is the displacement, µ is the shear modulus and κ = 3 − 4ν for plane strain and κ =

3−ν 1+ν

for

plane stress.

2.1.4

The Griffith Energy Balance

According to the first law of thermodynamics, when a system goes from a non-equilibrium state to equilibrium, there is a net decrease in energy [7]. In 1920, Griffith applied this idea to the formation of a crack [12]: ”It may be supposed, for the present purpose, that the crack is formed by the sudden annihilation of the tractions acting on its surface. At the instant following this operation, the strains, 8

and therefore the potential energy under consideration, have their original values; but in general, the new state is not one of equilibrium. If it is not a state of equilibrium, then, by the theorem of minimum potential energy, the potential energy is reduced by the attainment of equilibrium; if it is a state of equilibrium, the energy does not change.” The total energy must decrease or remain constant to form a crack or to allow its propagation. Thus the critical conditions for fracture can be defined as the point where the crack growth occurs under equilibrium conditions, with no net change in the total energy. Consider a wide plate subjected to a constant stress load with a crack 2a long (figure 2.4). In order for this crack to increase in size, sufficient potential energy must be available in the plate to overcome the surface energy γs of the material. The Griffith energy balance for an incremental increase in the crack area dA, under equilibrium conditions, can be expressed as follow:

dΠ dWs dΦ = + =0 dA dA dA

(2.13)

dΠ dWs =− dA dA

(2.14)

Or,

Where Φ is the total energy, Π the potential energy supplied by the internal strain energy and external forces and Ws the work required to create new surfaces.

Figure 2.4: A through-thickness crack in an infinitely wide plate subjected to a remote tensile stress [7] From Anderson [7], for the cracked plate illustrated in figure 2.4, Griffith [12] used the stress analysis of Inglis to show that, 9

Π = Π0 −

πσ 2 a2 B E

(2.15)

Where Π0 is the potential energy of an uncracked plate and B is the plate thickness. Since the formation of a crack requires the creation of two surfaces, Ws is given by

Ws = 4aBγs

(2.16)

With γs the surface energy of the material. Thus,

−

dΠ πσ 2 a = dA E

(2.17)

And,

dWs = 2γs dA

(2.18)

Solving for the fracture stress,

σf = (

2Eγs 1/2 ) πa

(2.19)

It is important to have in mind the distinction between crack area and surface area. The crack area is defined as the projected area of the crack (2aB in the present example), but since a crack includes two matching surfaces, the surface area is 2A .

2.1.5

The Energy Release Rate

Irwin [8], in 1956, proposed an energy approach for fracture that is essentially equivalent to the Griffith model, except that Irwin approach is in a form more convenient for solving engineering problems. Irwin defined an energy release rate G, which is a measure of the energy available for an increment of crack extension:

G=−

dΠ dA

(2.20)

The term rate, as it is used in this context, does not refer to a derivative with respect to time; G is the rate of change in potential energy with the crack area. Since G is obtained from the derivative of a potential, it is also called the crack extension force or the crack driving force. 10

According to equation 2.17, the energy release rate for a wide plate in plane stress with a crack of length 2a (figure 2.4) is given by,

G=

πσ 2 a E

(2.21)

Referring to the previous section, the crack extension occurs when G reaches a critical value,

Gc =

dWs = 2γs dA

(2.22)

Where Gc is also a measure of the fracture toughness of the material. At this moment, it must be said the energy release rate is extremely important for this thesis. This is justified by its direct relationship with the stress intensity factor. Irwin’s showed that for linear elastic materials, under loading Mode I, it may be written,

KI2 E0

(2.23)

E0 = E

(2.24)

G=

Where for plane stress,

And for plane strain,

E0 =

E 1 − ν2

(2.25)

Nevertheless, the energy release rate is still not enough and practical to get the value of the stress intensity factor.

2.1.6

The J-Integral

In the previous section, it was showed the basis behind the energy release rate, with a direct relation with the stress intensity factor. Even the energy release rate is a simple concept, it is not obvious how to deduce it with the finite element method. Fortunately, there is another concept in the LEFM theory, called the J-Integral, which may be calculated numerically and reveals itself very useful because in the context of LEFM, the J-Integral is equal to the energy release rate G. 11

J-Integral Calculation As said before, the stress intensity factor can be calculated by the energy release rate G, which in this thesis context is equal to the J-Integral. The J-Integral is a contour integral for bi-dimensional geometries (see figure 2.5). Its definition is easily extended to three-dimensional geometries, and it is used to extract the stress intensity factors.

Figure 2.5: a) 2D contour integral, b) 2D closed contour integral [6] For the two-dimensional case, the J-Integral is given by,

Z J = limΓ→0

n.H.qdΓ

(2.26)

Γ

Where Γ is the contour containing the crack tip, n is the exterior normal to the contour, and q is the unitary vector within the virtual displacement direction of the crack. The function H is defined by,

H = W I − σ.

∂u ∂x

(2.27)

Where W is the elastic strain energy1 , I the identity tensor, σ the stress tensor and u the vector of displacements. The contour Γ connects the two crack faces and encloses the crack tip. This is shown in figure 2.5 a). The contour tends to zero, until it only contains the crack tip (equation 2.26). The exterior normal n moves along the integration while q stands fixed in the crack tip. 1 The strain energy definition may also include the elastic-plastic effects, which are not presented, considering the fact that they will not be subject in this thesis.

12

It is very important to note the J-Integral is independent of the chosen path for elastic materials in the absence of imposed forces in the body or tension applied on the crack, so the contour does not need to contract itself on the crack, but it has only to enclose the crack tip. The two dimensional integral may be rewritten as a closed bi-dimensional contour integral as the following [13],

I

Z

J =−

m.H.¯ q dΓ − C+C− +C+ +Γ

t. C+ +C−

∂u .qdΓ ¯ ∂x

(2.28)

Where the line integrals are preformed in a closed contour, which is an extension of Γ. C+ and C− are contours along the crack faces, enclosed by C. The normal m had to be introduced as the unitary exterior normal to the contour C, respecting m = −n. The function q¯ had also to be introduced, being a unitary vector applied in the direction of the virtual extension of the crack tip, which respects q¯ = q in Γ and vanishes in C. In equation (2.28), t is the tension on the crack faces. Crack tension is also a subject not considered in this thesis. The second term may be erased from equation 2.28. The J-Integral may be now transformed in a surface integral by the divergence theorem properties, yielding to,

Z J=

( S

∂ ).(H .q)dS ¯ ∂x

(2.29)

Where S is the area in the closed domain. The equilibrium forces equation is,

(

∂ ).σ + f = 0 ∂x

(2.30)

Where σ is the tension tensor, and f the volume forces. And the energy strain gradient, for an homogeneous material with constant properties is,

(

∂W () ∂W ∂ ∂ )= =σ ∂x ∂ ∂x ∂x

(2.31)

Where  is the mechanical strain. Considering these two previous equations, the J-Integral may now be written as,

J =−

 Z  ∂ q¯ ∂u ( H + (f. ).¯ q )dS ∂x ∂x S

(2.32)

The bi-dimensional equation for the J-Integral is easily extended to a three dimensional formulation. The J-Integral has to be defined in order to a parametric variable s, in the crack front, in such manner 13

J(s) is defined by a function which characterizes the bi-dimensional J-Integral for each point placed in the path defining the crack front, which is also described parametrically in order to s (figure 2.6).

Figure 2.6: a) Local coordinates system, b) 3D J-Integral [6] The local system of cartesian coordinates is placed in the crack front. See figure 2.6 a). The axis x3 , runs tangentially the crack, x2 is defined perpendicular to the crack front. In this formulation, x1 will always be directed forward at the crack front. x1 and x2 define a perpendicular plane to the crack front. J(s) is so described in the x1 x2 plane. From figure 2.6, it is obvious that for the three-dimensional case, each infinitesimal 2D contour must be integrated, for each position of s, along the path described by the crack front in order to obtain a volume J-Integral.

Stress Intensity Factors Extraction

Having defined the procedure to obtain the J-Integral, for both, bi-dimensional and three-dimensional R crack geometries, it becomes necessary to extract the stress intensity factors. Consulting Abaqus

Documentation [6], for a linear elastic material, the J-Integral is related to the stress intensity factors by the following relation,

J=

1 T −1 K P K 8π

T

With K = [KI , KII , KIII ] and P the pre-logarithmic energy factor tensor [14, 15, 16, 17]. For homogeneous and isotropic materials this equation may be simplified in the form, 14

(2.33)

J=

1 3 1 2 (K 2 + KII )+ K E0 I 2G III

(2.34)

Where, E 0 is given by equations 2.24 and 2.25. At last, for pure Mode I loading, the relation between the J-Integral and the stress intensity factor is given by,

J =(

KI2 ) E0

Which is exactly the equation presented in section 2.1.5.

15

(2.35)

2.2 2.2.1

The Finite Element Method System of Equations

In this section, are presented the governing equations of the finite elements, used for the analyses.

Figure 2.7: FEM domain and boundary condtions [5] Considering the domain Ω of figure 2.7, the border Λ may be divided in four independent borders: Λt − with tension applied, Λu with imposed displacements, and the last two domains,Λ+ c and Λc , representing

the crack faces. The equilibrium equations and boundary conditions are,

∇σ + f = 0 in Ω

(2.36)

σ.n = t¯ in Λt

(2.37)

σ.n = 0 in Λ+ c

(2.38)

σ.n = 0 in Λ− c

(2.39)

u = uimp in Λu

(2.40)

Where f represents the volume forces, n the outer normal, t¯ the superficial forces, uimp the imposed displacements. Finally, considering an infinitesimal deformation δv, the weak formulation is,

Z

Z σ∇δv =

Ω

2.2.2

Z tδvdΓ +

Λt

f δvdΩ

(2.41)

Λ

Constitutive Relations

Although the fact the weak formulation has always the same form, the element quality depends on the constitutive relations, as well of the selected shape forms. This thesis is limited to the linear elastic fracture mechanics, implying that only small strains will be considered. The material model could not be different than the presented next, which is a limitation R imposed by the commercial software of analysis Abaqus , allowing only this model for XFEM aid.

16

Respecting the elasticity theory, the tension obeys to the following,

             

σxx





xx



       yy  σyy         zz  σzz    = D    2xy  σxy        2xz  σxz     2yz σyz

(2.42)

Being D given by,



1

  ν  1−ν   ν E(1 − ν)   1−ν D= (1 + ν)(1 − ν)   0    0  0

ν 1−ν

ν 1−ν

0

0

0

1

ν 1−ν

0

0

0

ν 1−ν

1

0

0

0

0

0

1−2ν 2(1−ν)

0

0

0

0

0

1−2ν 2(1−ν)

0

0

1−2ν 2(1−ν)

0

0

0

             

(2.43)

Where σij and ij are the tension and strain components, E the Young’s modulus, and ν the Poisson coefficient.

2.2.3

Element Types

R In Abaqus , the geometries under analysis can be modelled with two types of volumetric ele-

ments: the tetrahedral and hexahedral, which remains the isoparametric element most used for threedimensional elasticity [18]. R Abaqus admits two formulations of this element. The linear element of 8 nodes, identified as C3D8,

and the quadratic of 20 nodes, C3D20 (figure 2.8 a and b). At each node, for both elements, there are three degrees of freedom, corresponding to three possible displacements. Thus, the element of 8 nodes, has 24 degrees of freedom, a number three times lower than the 60 degrees of freedom of the element of 20 nodes [6].

Figure 2.8: Three dimensional elements, (a) 8 nodes, (b) 20 nodes, (c) 10 nodes, from [6] 17

As for the tetrahedral, there is a linear element of 4 nodes, C3D4 and a quadratic element with 10 nodes, the C3D10 (figure 2.8 c). Any of the four elements allows two kinds of numerical integration; reduced or full integration. The reduced integration is identified by a R in the element code. For example, C3D20R, indicates a threedimensional element of 20 nodes, being a quadratic with reduced integration.

18

2.3

The Classical Approach to the Stress Intensity Factor Calculation

In order to have a full understanding of the XFEM, it is necessary to evaluate the geometries with the classical approach for the stress intensity factor calculation. In the classical approach, according to [7], in two-dimensional problems quadrilateral elements are collapsed to triangles where three nodes occupy the same point in space, like what is shown on figure 2.9. For three dimensions problems, a brick element is degenerated to a wedge (figure 2.10).

Figure 2.9: Degeneration of a quadrilateral element into a triangle at the crack tip [7]

Figure 2.10: Degeneration of a brick element into a wedge [7] In elastic problems, the nodes at the crack tip are normally tied, and the mid-side nodes moved to the √ 1/4 points. This modification is necessary to introduce a 1/ r strain singularity in the element, which brings numerical accuracy due to the fact that the analytical solution contains the same term. A similar result can be achieved by moving the midside nodes to 1/4 points in non collapsed quadrilateral elements, but the singularity would exist only on the element edges; collapsed elements are preferable in this case because the singularity exists within the element as well as on the edges. When a plastic zone forms, the singularity no longer exists at the crack tip. Consequently, elastic singular elements are not appropriate for elastic-plastic analyses. Figure 2.11 shows an element that exhibits the desired strain singularity under fully plastic conditions. 19

√ Figure 2.11: Crack-tip elements for elastic and elastic-plastic analyses. Element (a) produces a 1/ r strain singularity, while (b) exhibits a 1/r strain singularity (a) Elastic singularity element and (b) plastic singularity element [7] According to [1], [6] and [7], for typical problems, the most efficient mesh design for the crack-tip region is the “spider-web” configuration (figure 2.12), consisting of concentric rings of quadrilateral elements that are focused toward the crack tip. The elements in the first ring are degenerated to triangles, as described above. Since the crack tip region contains stress and strain gradients, the mesh refinement should be greater at the crack-tip. The spider-web design allows a smooth transition from a fine mesh at the tip to a coarser. In addition, this configuration results in a series of smooth, concentric integration domains (contours) for the J-Integral calculation.

Figure 2.12: Spider-web mesh from [6]

20

2.4

The eXtended Finite Element Method

The extended finite element method, XFEM, is an evolution of the classical finite element method based on the concept of partition unit, i.e. the sum of shape functions is equal to one. This method was initially developed by Ted Belytschko [3] and his colleagues in 1999. The XFEM based on the concept of partition of unity [19], adds a priori known information about the solution of a given problem, to the FEM formulation, making possible, for example, to represent discontinuities and singularities, independently of the mesh. This particular feature makes this method very robust and attractive to simulate the propagation of cracks, since it is no longer necessary to have a continual updating of the mesh. The XFEM is then referenced as a Meshless method. In the XFEM, enrichment functions are added to additional nodes, in order to include information about discontinuities and singularities around the crack. These functions are the asymptotic near-tip solutions, which are sensitive to singularities, and the Jump function, which simulates the discontinuity when the crack opens.

2.4.1

XFEM Enrichment: Jump Function

To explain the form how the discontinuities are added to the FEM, consider a simple bi-dimensional geometry (figure 2.13), with four elements and an edge crack.

Figure 2.13: Bi-dimensional geometry for the XFEM deduction The solution for the displacement is typically given by,

u(x, y) =

10 X

Ni (x, y)ui

(2.44)

i=1

Where Ni (x, y) is the shape function on node i with coordinates (x, y), and ui is the displacement vector. 21

Defining c as a middle point between u9 and u10 and d as the distance between the two nodes, it is possible to write,

c=

u9 + u10 2

(2.45)

d=

u9 − u10 2

(2.46)

But also,

u9 = c + d

(2.47)

u10 = c − d

(2.48)

Manipulating the expression 2.44,

u(x, y) =

8 X

Ni (x, y)ui + c(N9 + N10 ) + d(N9 + N10 )H(x, y)

(2.49)

i=1

Where the Jump function was added obeying to,

H(x, y) =

  1

y>0

 −1

y